More than ten years after NCTM’s Curricu-lum and Evaluation Standards for SchoolMathematics (1989) first called for

reforms in mathematics education, many schoolsstill have not implemented those reforms. Wherethe Standards did take hold, they often failed tosurvive. As Principles and Standards for SchoolMathematics (NCTM 2000) notes, “Despite theconcerted effort of many classroom teachers,administrators, teacher-leaders, curriculum devel-opers, teacher educators, mathematicians, andpolicymakers, the portrayal of mathematics teach-ing and learning in Principles and Standards isnot the reality in the vast majority of classrooms,schools, and districts” (p. 5).

This article looks at the past reform movementin mathematics from the perspective of a classroomteacher who has successfully refocused the class-room curriculum around problem solving, reason-ing and proof, communication, connections, andrepresentation. Although this article primarily willexamine only one component of the reformagenda, this component is central to the reformmovement. It involves moving mathematicsinstruction away from teaching mathematicsthrough drill and practice and toward teachingmathematics through problem solving.

I base my comments on my own classroomexperiences and the experiences of other teacherswho have found that problem solving can be diffi-

cult to teach and even more difficult for students tolearn. First, I examine why implementing the pastNCTM Standards in their classrooms was so diffi-cult for some teachers. Second, I offer suggestionsto help teachers implement the revised Standards.

What Went WrongWhen faced with the expectation that they teachproblem solving, many teachers unfortunatelyengage in what I call the “blame game.” This reac-tion can include four stages and consists of search-ing for someone or something to blame for the dif-ficulties that teachers encounter when attemptingproblem solving with children. Teachers must beaware of these potential challenges because some-times knowing what to avoid is as important asknowing what to embrace.

Stage one of the blame gameThe first stage involves blaming yourself. Manyteachers become convinced that their lack ofknowledge is the source of their difficulties. Typi-cal comments that teachers make at this stageinclude the following:

• “I could never figure out story problems when Iwas in school; how can I teach students how todo them?”

• “I’ve tried to teach problem solving, but itseems like I always end up showing the childrenwhat to do and it turns into just another type ofdrill-and-practice exercise.”

Teachers often try to expand their knowledge bytaking college classes or attending workshops on

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Larry Buschman, buschmanlf@proaxis.com, teaches a multiage class (grades 1–3) at JeffersonElementary School in Jefferson, Oregon. He is interested in helping children become confidentand capable problem solvers in mathematics.

By Larry Buschman

Teaching Problem Solving in Mathematics

Copyright © 2004 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

how to teach children problem-solving heuristics(look at the problem, choose a strategy, solve theproblem, look back) or problem-solving strategies

(guess and check, make a model, look for patterns,estimate, and so on). Although this type of profes-sional development can be helpful, good teachers

Teaching Children Mathematics / February 2004 303

Figure 1Children solving problems in ways that make sense to them

The problem:Hillary is 52 inches tall and Chloe is 39 inches tall. How much taller is Hillary than Chloe?

Below are typical children’s solutions. Although some of the strategies may seem unorthodox and inef-ficient, they represent children’s genuine attempts to make sense of the problem and communicate theirsolutions using a drawing, a manipulative, an oral description, or a computation procedure.

Jonathan (first grade):Jonathan first drew 41 lines to represent Chloe’s height ininches. He then recounted the lines and discovered that hehad drawn 41 instead of 39, so he crossed out 2 lines. Next,he drew 52 lines to represent Hillary’s height. He then pro-ceeded to cross out one line at a time on Chloe’s row andon Hillary’s row. He continued this process until he hadcrossed out all the lines on Chloe’s row. He then countedthe remaining lines on Hillary’s row and wrote 13 for hisanswer.

James (first grade):James used a popular classroom manipulative. He made a row of 39 tiles on the floor “’cause theywouldn’t fit on my desk.” Then he made a row of 52 tiles and counted “the ones that were more ’causethat’s how much more she is.”

Jose (second grade):Jose used a different manipulative than did James—his fingers. Jose put 39 in his head and then usedhis fingers to count on from 39 to 52. He paused as he reached 49 so he could put “10 in my head forlater.” When he reached 52, he held up 13 fingers, so he said, “She is 13 inches more big.”

Natasha (second grade):Natasha used an approach similar to the direct modelingmethod that Jonathan used, but she has less of a chance ofmaking a counting error because she used bars to representthe quantity 10. Also, her approach might be more accuratethan the counting-on method used by Jose because shedoes not have to keep track of quantities in her head.

Brandi (third grade):“52 is 40 + 12 and 39 is 40 – 1 so it’s 12 up and 1 down and that makes 13.”

Brandi used an approach common to some children: a familiar reference point from which to work for-ward and backward in order to facilitate her computations. In this case, she chose to use a multiple of10 so she could “solve the problem in my head.”

Eli (third grade):“52 is 4 inches more than 4 feet and 39 is 3 inches morethan 3 feet, so minus them and you get 1 inch and 1 footand that’s 13 inches.”

of problem solving do not necessarily need to beexpert problem solvers themselves. Instead ofteaching children how to solve certain types ofproblems using traditional problem-solving strate-gies, teachers can encourage children to solveproblems in ways that make sense to them (see

fig. 1). Engaging in sense making while solvingproblems is one way that children deepen theirunderstanding of mathematics.

Once children have solved problems, they canshare their personal strategies with classmatesand respond to questions or comments that theirpeers and teacher make. Putting their solutionprocesses into words is a second way that childrendeepen their understanding of mathematics. Whenchildren write or talk about how they solved aproblem, they must organize and clarify theirthinking as they attempt to communicate theirideas clearly and completely. Finally, children cancompare all the different solutions that their class-mates have shared and examine how they are thesame or different. Reflecting on how differentways of solving problems are related or connectedis a third way children deepen their understandingof mathematics.

Stage two of the blame game The second stage in the blame game involves blam-ing the mathematics problems. Typical commentsthat teachers make at this stage include the following:

• “If someone would just write some good prob-lems, it wouldn’t be so bad.”

• “Sometimes the problems are so wordy, it’s morelike a reading test than a mathematics activity.”

Teachers at this stage frequently ask, “Who haswritten the book of good problems, and where can Ibuy a copy?” However, there usually is nothingwrong with the problems that teachers already use.Although using “good” problems is important, thequality of the problems does not necessarily deter-mine the success or failure of a problem-solvingprogram. In fact, some of the best problems for chil-dren to solve exist in the day-to-day activities thatoccur in the classroom. To take advantage of theseproblems, teachers must listen to what children sayand do, and then help them become aware of themathematics embedded in these events (see fig. 2).Teachers can construct problems out of the mathe-matical data that surrounds children as they partici-pate in lunch count, share classroom and playgroundequipment, construct objects with blocks, createdesigns with pattern blocks, or explore Internet sites.

Stage three of the blame gameThe third stage of the blame game involves blamingthe students. Out of desperation, teachers makecomments such as the following:

304 Teaching Children Mathematics / February 2004

Figure 2An example of a problem based on a classroom event

Young children love to pose “What if . . . ?” questions in an attempt to stumpadults. One day a child posed the following question: “What if you didn’thave rulers? How could you measure stuff?” The next day, I presented thisproblem for the all the children in the class to solve:

You need a straw that is 2 inches long, but you don’t have a ruler. Youhave only a straw that is 3 inches long and a straw that is 5 inches long.Using only the 3-inch and 5-inch straws, how could you make a straw thatis 2 inches long?

One advantage of this problem is that after children have solved it, they canuse real straws and verify their solution.

Most adults probably would solve this problem by placing the two strawsside by side, lining up the straws at one end, and then cutting off the sectionof the 5-inch straw that extends beyond the end of the 3-inch straw because5 – 3 = 2.

Interestingly, only a few children chose to solve this problem in the mannershown above.

Kohl (first grade):“Glue the 5 and the 3 sticks together at the ends and break it in half, andthen break it in half again and it makes 2 long.”

Todd (second grade):“I know if you put 3 dimes beside each other it is almost 2 inches, but it isjust a little bigger, ’cause Zachary and I measured it the other day when wewere doing an experiment. And 2 quarters is almost 2 inches but it is just alittle smaller. So 2 inches is in the middle between the dimes and the quar-ters, so make the straw this long.”

Hillary (third grade):“Divide the 3-inch one into thirds and make the marks on the straw. Counttwo of them [the marks] and cut it ’cause that’s 2 inches.”

2 inches

3 inches

5 inches

• “My students just don’t try. Unless I tell themwhat to do and how to do it, they just sit andwait for me to do it for them.”

• “Children are so impatient. Unless the problemcan be solved in five seconds or less, they don’twant to have anything to do with it.”

Fortunately, their students’ lack of particularskills or abilities is not the reason that mostteachers have so much difficulty teaching prob-lem solving. Children appear to be natural prob-lem solvers, and problem solving is one of theprimary functions of the human brain (Jensen1998; Sylwester 1995). Although children fre-quently solve problems in unusual or confusingways, their solutions often reflect the ingenuityand creativity of young minds (see fig. 3). Whenchildren are given the chance to solve problemsin ways that make sense to them, they are capa-ble of solving problems often considered beyondtheir abilities by using their informal mathemati-cal knowledge to—

• take ownership of the problem and craft a solu-tion from a personal perspective;

• develop unique strategies for solving a widerange of problems;

• construct meaningful connections between whatthey know and what they are learning; and

• grow as capable problem solvers who can solveproblems with confidence, communicate theirstrategies orally and in writing, and reflect onthe solutions of others.

Children can not only solve problems but alsowrite their own problems (see fig. 4). Students thensolve one another’s problems and the authors of theproblems agree or disagree with each solution.These problems serve a dual purpose: They givechildren a chance to take control of the mathemat-ics curriculum and they give teachers a way toexamine what children think are the characteristicsof good problems.

Stage four of the blame gameThe final stage of the blame game becomes theturning point for many teachers. Either they passthrough this stage and go on to become success-ful teachers of problem solving or, after blamingeveryone and everything, they simply give up andleave this difficult task to someone else. In thisfinal stage, teachers blame “those other people”and make comments that include the following:

• “If the district curriculum specialists wouldleave us alone and let us teach mathematics, wewouldn’t have all these problems.”

• “How can I be expected to teach problem solv-ing along with all the other things I am sup-posed to cover in the state-mandated mathemat-ics curriculum?”

The reason that many teachers feel frustrated intheir attempt to teach problem solving, however, isnot that individuals outside the classroom areimposing unreasonable expectations on them andtheir students. In fact, the world outside the class-room is full of problems that need to be solved.Many state departments of education support theteaching of problem solving and include thesetypes of questions on state tests (see fig. 5).

If teachers are not the problem, and the mathe-matics problems are not the problem, and the stu-dents are not the problem, and “those other people”are not the problem, why is problem solving so dif-ficult to teach and even more difficult for studentsto learn? A very simple answer to this questionexists: Teaching problem solving using the tradi-tional, drill-and-practice model of mathematicsinstruction is difficult. Although this type ofinstructional model is effective in helping childrenremember mathematics facts and computation pro-cedures, it is less effective in helping children learnhow to become problem solvers.

Problem solving requires understanding thatgoes beyond memorized facts and computationalgorithms. Most teachers who work with youngchildren eventually realize that although they cantransfer information to young learners, transferringunderstanding is very difficult. This is becauseeach individual crafts his or her understanding asinformation is transformed into useful and usableknowledge. Problem solving requires the ability touse information purposefully, not just the ability tomemorize and remember it.

The Challenges ThatTeachers of ProblemSolving Face

Teaching problem solving poses many challengesfor teachers. First, teachers must change not onlywhat they teach but also how they teach. Thesechanges in methodology must extend beyondfine-tuning existing instructional practices; teach-ers must adopt fundamentally different ways of

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teaching and assessing children. In addition,teachers must acquire subject-matter knowledgethat is deeper and more flexible than that requiredto follow a lesson plan or directions in a teacher’smanual. As teachers of mathematics facts and for-mulas, teachers must transfer what they know totheir students. As teachers of problem solving,however, their role shifts to helping studentsextend and expand what they already know inorder to learn even more. Teachers can do this byusing children’s original solutions to problems asa basis for classroom conversations and thenensuring that the students discuss significantmathematics.

A second challenge is that teachers must modelfor students problem-solving abilities that they nei-ther possess nor have seen demonstrated by others.In short, the NCTM Standards ask teachers to beteachers that their professional development hasnot prepared them to be and that they have notexperienced in their own lives as learners. One waythat teachers can learn to become successful teach-ers of problem solving (and students can learn tobecome problem solvers) is to spend time in thecompany of others who can—

• model appropriate behaviors, dispositions, andskills;

• engage the novice in an ongoing conversationwithin a community of learners; and

• give the novice timely feedback.

Unfortunately, these models and learning commu-nities do not exist in many of the universities whereteachers are trained or in the schools where theyteach. The good news is that these models arebecoming available as more teachers incorporateproblem solving into the curriculum.

A third challenge for teachers of problem solv-ing is that they may need to change some of theirmost basic beliefs about what constitutes mathe-matical literacy. At the heart of these beliefs is thevery definition of mathematical literacy:

• Do children demonstrate mathematical literacywhen they memorize facts and computationalgorithms? Is problem solving the ability toidentify which computation procedure will findthe answer to story problems?

• Or do children demonstrate mathematical literacyby using mathematical information purposefullyto build relationships between the big ideas inmathematics? Is problem solving the ability tosolve a wide range of problems even when thesolution to the problem is not immediatelyknown?

306 Teaching Children Mathematics / February 2004

Figure 3Impossible problems

Problem solving often requires creative as well as critical thinking. In order to stretch children’s thinking,I sometimes pose problems that encourage children to “play with ideas” and “think outside the box.”These problems have earned the nickname “impossible problems” because they appear to lack a math-ematical solution.

How can two children share 5 pennies?Zachary: “You could cut one of the pennies in half and they would each get 2 1/2 cents, but if you cut

money in half, it isn’t worth anything. So I don’t think you can do this problem.”Chloe: “They could each get 2 so it would be fair, and put the other penny in the bank until they got

another 5 cents, and then take the penny out and they each get 3 more cents.”Ariel: “They could buy a candy bar that costs 5 cents and break it in half, and then they each get half

a candy bar and it’s a fair share.”Eli: “They could buy a lottery ticket and when they win they could split the $1,000 prize, and they

would each get $500 and they would be rich forever.”

How can three children share 10 balloons?Benjamin: “Well, it’s impossible because if you try to cut a balloon in half it will pop.”Andrea: “They each take 3 and give the other balloon to their teacher because she needs to be

happy too.”Austin: “When no one is looking, let one of the balloons go and say, ‘Oops, it just slipped,’ and then

share the others so they each get 3 balloons.”Timothy: “Sell the balloons for 30 cents and share the money so they each get 10 cents.”

The fourth, and perhaps most difficult, challengeis that teachers may need to convince parents andothers that the changes that the NCTM Standardscall for are necessary and will significantly raisestudent performance.

ConclusionIn order to realize the vision of the NCTM Stan-dards for mathematics instruction, teachers needsupport and encouragement. Both students andteachers must learn new roles, behaviors, and

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Figure 4Student-authored problems with their classmates’ solutions

Brandi’s problem: My Mom baked 4 dozen cookies for the bake sale. If there are 6 cookies in each bagand if one person bought 3 bags for $3.00, how many dozen cookies were left?

Austin’s Solution Andrea’s Solution Todd’s Solution

Zachery’s problem: When we went out for pizza, my dad ordered 2 pizzas. My parents ate 3 pieces,each of them, and my brother and sister and I each ate 2 pieces. How many pieces of pizza were left?This is how they cut them.

Hillary’s Solution Brandon’s Solution Chloe’s Solution

skills. Learning to solve problems and learning tomemorize information require profoundly differ-ent ways of thinking and behaving. Instead ofembedding the teaching of traditional mathemat-ics skills in the context of problems, many teach-ers will continue to try to teach the traditional con-tent through drill and practice along with aseparate problem-solving curriculum and find thetask to be overwhelming.

Teachers need time to learn new content, time tolearn old content in greater depth, and time to con-struct personal understanding. Simply learningnew mathematics content is not enough; teachersmust also learn how to present the content so thatchildren learn mathematics with understanding.Teachers must start asking the right questionsbecause the questions they ask will determine theanswers they uncover. Instead of asking, “What isthe most efficient way to teach mathematics con-tent?” teachers should ask, “What is the mosteffective way for children to learn this content?”Although these two questions may seem similar,they yield surprisingly different answers. The firstquestion focuses on the quantity of information toteach and the second focuses on the quality of thelearning process.

Teachers also need professional developmentthat includes continuing education at the univer-

sity level, as well as the opportunity to attend con-ferences and workshops that introduce them to dif-ferent ways of teaching mathematics. Based on myexperiences as a workshop presenter, however, Irealize that the traditional one-shot workshop willnot result in the desired changes. Teachers mustvisit classrooms where problem solving is beingtaught successfully. Teachers need to see for them-selves that the problem-solving approach willwork in another ordinary classroom before theybelieve that it can work in their own classrooms.Conversely, many teachers cannot think about howto implement a problem-solving approach in theirclassrooms until they believe that it is actuallypossible.

Finally, teachers need a class size that is appro-priate to facilitate student understanding of mathe-matics. If we define learning in terms of memoriz-ing facts and algorithms through drill and practice,class size is somewhat irrelevant; thirty children cansit at their desks and complete arithmetic work-sheets as easily as ten students. If, however, wedefine learning in terms of using mathematics inpurposeful ways, class size is very relevant and thegreat number of students in many classroomsseverely limits the construction of understanding byall children.

Final ThoughtIn most school districts, the responsibility for orga-nizing, budgeting, and scheduling staff develop-ment belongs to the administrators of the district.Many administrators, however, lack an understand-ing of NCTM’s Standards and the type of staffdevelopment activities that would most benefitteachers. Therefore, these administrators may needto step out of their traditional oversight role andparticipate more directly in the staff developmentactivities of their schools or districts. BecauseNCTM’s Standards call for an entirely new way ofteaching and learning mathematics, administratorsmay need to move away from administering staffdevelopment activities and toward leading theirstaffs with insight. Teachers may need to do thefollowing:

• They should encourage school and districtadministrators to attend training sessions onproblem solving, visit classrooms where prob-lem solving is being taught successfully, andjoin with teachers to build a shared vision ofmathematics instruction.

308 Teaching Children Mathematics / February 2004

Figure 5Problem-solving questions similar to

those on a state test for third grade

1. Mr. B. lit a brand-new candle. After threehours, the candle had one-fourth of the waxremaining. How long will it take for the candleto burn completely?

2. Jason bought these things. If he gave theclerk a five dollar bill, what coins and bills willthe clerk probably use to give Jason hischange?

2 note pads: $0.35 each1 pen: $0.79 each1 ruler: $1.25 each

3. Which game did the Braves lose by themost points?

Day Braves MarinersTuesday 6 3

Friday 4 6

Saturday 2 3

• They should ensure that school and districtadministrators schedule staff developmentactivities over an extended period of timebecause most teachers will need severalyears to become confident and competentteachers of problem solving. Teachers willneed ongoing support and training to over-come the difficulties that they will face intheir classrooms.

• They should help school and district adminis-trators become knowledgeable leaders who canbuild local consensus in the school, district,and community about NCTM’s Standards anda problem-solving approach to learning mathe-matics. Any deviation from the traditionalapproach to teaching mathematics frequentlyproduces a strong backlash from some schoolboard members, parents, or politicians. Teach-ers and administrators must be prepared toaddress the concerns of the critics.

If teachers and administrators fail to fullyunderstand the magnitude of the changes thatNCTM’s Standards call for, reaching consensusin any school or district about what studentsshould know and be able to do as mathematicianswill be difficult. Some high school and middleschool teachers will continue to blame teachersof the grade levels below theirs for not ade-quately preparing students for the challenges oftheir classrooms. Similarly, some elementaryschool teachers will continue to blame parentsfor not preparing children for school and someparents will continue to blame schools for notteaching children mathematics the way that theywere taught. The blame game must stop. If edu-cators cannot reach a consensus, the “MathWars” that have been fought in California, Ore-gon, and elsewhere will continue to weaken thereform movement well into the future.

ReferencesJensen, Eric. Teaching with the Brain In Mind. Alexan-

dria, Va.: Association for Supervision and CurriculumDevelopment, 1998.

National Council of Teachers of Mathematics (NCTM).Curriculum and Evaluation Standards for SchoolMathematics. Reston, Va.: NCTM, 1989.

———. Principles and Standards for School Mathemat-ics. Reston, Va.: NCTM, 2000.

Sylwester, Robert. A Celebration of Neurons: An Educa-tor’s Guide to the Human Brain. Alexandria, Va.:Association for Supervision and Curriculum Devel-opment, 1995. ▲

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